Datig & Schlurmann - Performance and Limitations of the Hilbert-Huang Transformation (HHT) With an...

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Ocean Engineering 31 (2004) 1783–1834

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Performance and limitations of the Hilbert–Huang transformation (HHT) with anapplication to irregular water waves

Marcus Datig, Torsten Schlurmann �

Hydraulic Engineering Section, Civil Engineering Department, Bergische University Wuppertal,

Pauluskirchstr. 7, 42285 Wuppertal, Germany

Received 26 October 2003; accepted 18 March 2004

Abstract

This paper relates to the newly developed Hilbert–Huang transformation (HHT). Anoverview of this time-frequency analysis technique and its applications are given. Keyelements of the numerical procedure and principles of the Hilbert transformation (HT) areestablished. A simple parameter study with trigonometric functions to get an idea aboutthe numerical performance of the empirical mode decomposition (EMD) is performed. Themain results of estimating relative standardized errors made between analytically exactdefined sine waves and disintegrated intrinsic functions as well as their specific influence oneach other are determined. Practical applications are carried out next to evaluate computednonlinear irregular water waves based on Stokes perturbation expansion approach and mea-surements on fully nonlinear irregular water waves recorded in a laboratory wave flume.Correspondence between simulated and recorded wave trains is given for narrow-bandedfundamental components. Deviations are unveiled when carrier and riding waves get broadbanded. Time-dependent spectral representation shows signs of an interesting phenomenonas instantaneous frequencies and amplitudes exhibit strong correlations with water surfaceelevations of both numerical and measured data series.# 2004 Elsevier Ltd. All rights reserved.

Keywords: Time-frequency analysis techniques; Hilbert–Huang transformation; Empirical mode

decomposition; Irregular water waves; Perturbation expansion approach; Irregular second order Stokes

wave theory

M. Datig, T. Schlurmann / Ocean Engineering 31 (2004) 1783–18341784

1. Preliminaries

Time-frequency dependent analysis methods are a novel approaches in the fieldof applied mathematics and have rapidly emerged as a common subjects ofresearch and application in scientific and engineering investigations in the last twodecades (Meyer, 1993). Consequently, these techniques are also applied to ordinarywave data and are approved in the coastal and ocean engineering disciplines.Liu (2000a–c) introduces the wavelet transform technique in its entirety and, conse-quently, analyzes coastal wave data sets. Chien et al. (1999) and Schlurmann (2000,2002) investigate extreme wave events—transient water waves with episodicextreme wave heights—around the island of Taiwan and in the Sea of Japan withthe wavelet transformation and the newly developed Hilbert–Huang transform-ation (HHT) (Huang et al., 1998, 1999), respectively, and conclude that time-vari-ant frequency analysis methods are more appropriate mathematical tools ofexamining transient water wave phenomenon than ordinary, time-invariant Four-ier-based techniques. Although these simple routines have become the most valu-able tool in spectral data analysis today, some regulatory principles have to beaccomplished. For example, Titchmarsh (1948) points out that Fourier transforma-tions are rigorously restricted to linear systems and stationary data series, so thatdata series can be mathematically characterized from a rather global point of viewin a general form of a finite number of time-invariant sinusoidal operators eixt. Thesignal is therefore transformed and represented only in the frequency domain.Information of time is completely abandoned, so that transient phenomena ortime-variant characteristics and modulations are cancelled out in Fourier represen-tation (see e.g. Schlurmann, 2002). To overcome these limitations, several time-variant methods, e.g. windowed-Fourier transforms or Wigner–Vile techniques(Huang et al., 1998, 1999) as well as the wavelet based approaches, have beendeveloped and applied in several engineering disciplines. However, it is not in thescope of the present paper to review those particular methods in depth and derivesubsequent performance tests for each.The paper is structured as follows: we will first give a general overview on the

HHT technique. The numerical procedure of the empirical mode decomposition(EMD) is made clear and the principles of the Hilbert transformation (HT) areintroduced in Section 3. In Section 4, importance is laid on the cubic spline interp-olation, especially on the algorithms including their theoretical background, sincethis feature plays an essential part when decomposing original data series intointrinsic mode function (IMF). Furthermore, the end conditions of the interp-olation routines are significant, besides showing how it could be possible to fix thecubic splines to additional characteristic points of the data series to get the bestpossible approximation within the iterative algorithm. Next, a pragmatic investi-gation done within a principle parameter study with simple trigonometric functionsis presented in Section 5 to get an idea about the numerical performance of theEMD. The principle results are shown to estimate relative standardized error madebetween analytically exact defined sine waves and disintegrated intrinsic functionsand their specific influence among each other. Subsequently, limitations and func-

1785M. Datig, T. Schlurmann / Ocean Engineering 31 (2004) 1783–1834

tional boundaries of the EMD are derived from this analysis. Finally, Section 6presents practical applications to theoretically evaluate calculated nonlinear irregu-lar water waves with measurements of fully nonlinear irregular water wavesrecorded in a laboratory wave flume. The results and summary are provided anddiscussion is initiated in the final section of the paper.

2. Introduction

The present investigation takes up the recently developed HHT as its focus. Thistechnique is expected to decompose time-dependent data series into its individualcharacteristic oscillations with the so-called EMD. This procedure is capable ofempirically disintegrating any complex set of data into a finite number of hiddenembedded IMF. These functions allow the calculation of a multicomponentinstantaneous frequency representation that admits well behaved Hilbert trans-forms. The EMD method was developed to operate on the data being processedand prepare them for the Hilbert transform. Applying the Hilbert transform toeach of these extracted IMFs reveals the time-variant frequency and amplitudeHilbert spectrum.Practical applications of the HHT are today broadly spread in numerous scien-

tific disciplines and investigations, e.g. on gravity wave characteristics in the middleatmosphere (Zhu et al., 1997) to derive useful physical insights into dispersive–dis-sipative wave phenomenon and on the ages of large amplitude coastal seiches onthe Caribbean coast (Huang et al., 2000). Further, Tabor et al. (submitted for pub-lication) make an effort to figure out the relationship between tropical surface tem-perature oscillations and vegetation dynamics in Northern Brazil during 1981–2000and Pinzon (2002) successfully uncouples seasonal and interannual components inremotely sensed data with the HHT to identify and remove solar zenith angletrends from normalized difference vegetation index time series. Similar investiga-tions are performed in analyzing the monthly Southern oscillation index (SOI) inorder to derive cleaner representation and allow improved forecasting of El Nino/Southern oscillation (ENSO) dynamics (Salisbury and Wimbush, 2002). Moreover,the HHT has been used in other fields of geophysics, e.g. to examine earthquakeprocesses as well as for the determination of the dispersion curves of seismic sur-face waves (Magrin-Chagnolleau and Baraniuk, 1999; Huang et al., 2001; Chenet al., 2002; Oonincx, 2002), and to study the effects of seismic motions on the con-dition of buildings and structures in civil engineering (Vincent et al., 1999; Salvinoand Pines, 2001). Additionally, the HHT is used in tsunami research to detectearthquake generated water waves from data series recorded from bottom pressuretransducers in the Northern Pacific (Schlurmann et al., 2001) and to examine theresponses of New Zealand coastal waters to the Peru tsunami of 23 June 2001(Goring, 2002).Some work on the HHT has also been performed in medical sciences, e.g. in

achieving artifact reduction in electrogastrograms due to the fact that severe con-tamination effects take place (Liang et al., 2001), and, in disintegrating multisite


neuronal data (Hofmann et al., 2001). Kuchi and Koch (2002) make use of theEMD in automatic human gait analysis that is becoming increasingly importantin the context of human gesture recognition to serve as an individual biometriccharacteristic. However, on the whole, practical assessments on the usage of theHHT have been preliminary done in the coastal and ocean engineering dis-ciplines: e.g. in an investigation on an automated routine to detect breakingwaves from records in deep water environment under laboratory conditions (Zim-mermann and Seymour, 2002), and in an assessment of the fatigue analysis ofmarine risers which have multiple excitation responses that exhibit an intermittentbehaviour at multiple frequencies (Gravier et al., 2002). Veltcheva (2002) uses theHHT to wave field data from nearshore area to study their group structure aswell as the shoaling characteristics and cross-shore variations along the beachprofile. Another study done by Schlurmann et al. (2002) considers the hydro-dynamic wave damping performance of an artificial reef in which the HHT ischosen to describe the physical processes of the transmitted transient wavesinduced by the presence of the submerged structure. It is proven that non-station-ary sea states behind the reef are essentially composed of individual dispersivewaves which are phase shifted and propagate individually to generate a kind ofbeat effect. This dynamic behaviour is effectively described by the HHT. More-over, when we regard real sea states or any other ‘‘real-world signal’’ character-ized by distinct nonlinear and non-stationary individual pattern, the tentativeneed to improve the existing analyzing spectral techniques is felt. From Huang’spoint of view, attempts to solve the underlying partial differential equations, aswell as the Fourier-based analysis methods are both generally based on the per-turbation expansion approach, i.e., reducing a set of nonlinear equations to asuperposed system of linear ones. But the solution obtained by the perturbationexpansion approach often makes little physical sense, since the properties of anonlinear equation system should definitely be different from a simple collectionof linear ones. These facts can be regarded as some of the reasons to intensifyfurther exploration concerning time-frequency analysis methods as the Hilbert orwavelet transformation are.In a few recent theoretically based studies on the HHT, investigations on

improving the iterative sifting procedure with non-uniform sampling intervals(Meeson, 2001) or on using the mean value theorem for detecting further charac-teristic points in the data series (Qiang et al., 2001) were carried out. An excellentand innovative application of the HHT was presented by Long et al. (2002). Theyexpanded this time-variant frequency analysis method to include the analysis ofimage data in two dimensions. In this context, it is used to identify water waveslopes under the action of high wind and in the presence of lower frequency back-ground waves. Nevertheless, little work has been carried out and verified about theperformance, restrictions and limitations of the HHT; except what has been pub-lished yet in both introductory papers from Huang et al. (1998, 1999). In accord-ance to this lack of clarity, the current paper attempts to close this gap, andconsequently, introduces some pragmatic benchmark tests as well as comprehen-sive solutions on the numerical performance, stringent limitations and the func-


tional boundaries of the HHT. Validations are carried out with an application tolaboratory generated nonlinear water waves. On the whole, the present investi-gation has been conducted within the framework of a research project funded bythe German Science Foundation [DFG, SCHL 503/5-1].

3. The Hilbert–Huang transformation

3.1. Empirical mode decomposition

As introduced in detail by Huang et al. (1998, 1999), the patented technique ofthe EMD is capable of adaptively decomposing signals into oscillating intrinsiccomponents. The underlying source to derive IMF from the data series is based onand derived from the data series itself. The key innovation embodied in the methodis the introduction of the IMF, which is based on local natural properties of thesignal and which gives a subsequent meaning to the concept of instantaneous fre-quency which will be explained in the following. An IMF can be best defined as ahidden oscillation mode that is embedded in the data series, since it is allowed tobe non-stationary and either be amplitude and/or frequency modulated. Accordingto Huang et al. (1998, 1999), an IMF is defined as a function that satisfies thefollowing two conditions:

. The number of extrema and thus the number of zero-crossings in the whole dataseries must be equal or differ at the most by one.

. At any instant in time, the mean value of the envelope defined by the local max-ima and the envelope of the local minima is zero.

The first condition is similar to the narrow-band requirement for a stationaryGaussian process. It ensures that the local maxima of the data series are alwayspositive and the local minima are negative, respectively. The second conditionmodifies a global requirement to a local one, and is necessary to ensure that theinstantaneous frequency will not have unwanted fluctuations as induced by asym-metric waveforms. Regarding an arbitrary data series x(t), the IMFs are obtained,using the following algorithm, shown by Schlurmann (2002):

(1) I
nitialize: r0ðtÞ ¼ xðtÞ, i ¼ 1 (2) E xtract the ith IMF:
(a) Initialize: h0ðtÞ ¼ riðtÞ, k ¼ 1(b) Extract the local maxima and minima of hk�1ðtÞ(c) Interpolate the local maxima and the local minima by a cubic spline toform upper and lower envelopes of hk�1ðtÞ(d) Calculate the mean mk�1ðtÞ of the upper and lower envelopes of hk�1ðtÞ(e) Define: hkðtÞ ¼ hk�1ðtÞ �mk�1ðtÞ(f) If IMF criteria are satisfied, then set IMFiðtÞ ¼ hkðtÞ else go to (b) withk ¼ k þ 1


Define: riðtÞ ¼ ri�1ðtÞ � IMFiðtÞ
(3)(4) If riðtÞ still has at least two extrema, then go to (2) with i ¼ i þ 1; else thedecomposition is completed and riðtÞ is the ‘‘residue’’ of x(t).
At the beginning, the original data set x(t) is initialized as r0(t). This initializa-tion can be characterized as the introduction to the outer loop to decompose theinput signal into successive IMFs. The second inner loop is started to find everysingle IMF. Again this loop is initiated by an introductory process, setting ri as thestarting array for the inner loop. The first run of the inner loop, array hk�1ðtÞ withk ¼ 1, corresponds to the original data series x(t). Extrema of the signal arerevealed next. The minima and maxima are linked by a cubic spline to form anupper and lower envelope of x(t). Then the corresponding mean mk�1ðtÞ is definedas the difference of upper and lower envelopes and subtracted from the initial dataseries hk�1ðtÞ to represent a tentative first IMF hk(t). The conditions of defining anIMF are subsequently approved. Usually, after the first run, the criteria are notsatisfied; so the inner loop is restarted from (b) by using hk(t) to initialize hk�1ðtÞwith k ¼ k þ 1. This so-called ‘‘sifting’’ process is repeated until the stopping cri-teria are fulfilled. Then the first IMF is disintegrated and the whole procedure isredone to sift additional IMF from the data series x(t) provided that the stoppingcriteria of the outer loop fail. The iterative decomposition process ends when thestopping criteria of the outer loop are satisfied so that r(t) is taken as the residueof x(t) which can also be interpreted as the trend of the signal. The original signalx(t) is then represented through the sum of a specified number of IMFs so that

xðtÞ ¼Xni¼1

IMFi þ rðtÞ ð1Þ

where n is the total number of IMFs and r(t) is the residue of the sifting process.Due to this iterative procedure, none of these sifted IMFs is derived in the closedanalytical form. Another way of explaining how the EMD works is that it picksout the highest frequency oscillation that remains in the signal. Thus, locally, eachIMF contains lower frequency components than the one extracted just before.Therefore, the EMD is based on the direct extraction of energy associated withvarious intrinsic time scales. This property can be very useful to detect rapid fre-quency changes, since a change will appear even more clearly at the level of anIMF. The extensive literature review on the practical applications of the HHTgiven in Section 2 shows that as a rule of thumb the average number of IMFsderived from natural data series varies from about 6 up to 10 (12).Completing this decomposition method is technically assured and is only depen-

dent on the precision of the numerical sifting process. Mathematical orthogonalityof the set of IMFs is not guaranteed in the theoretical sense that hIMFi; IMFji ¼ 0for i 6¼ j, since these intrinsic functions are not given in a closed analytical form.But this condition is satisfied in an empirical way, meaning that two IMFs arepractically orthogonal within a sufficient diminutive interval at a certain instant in


time (Huang et al., 1998, 1999). Moreover, it is noteworthy that the derived IMFsdo not guarantee a well-defined physical meaning and great caution is advised whenattempting to interpret them.

3.2. Hilbert transform

The Hilbert transform (HT), and its general properties, when applied to theanalysis of linear and nonlinear system, are discussed at great length by Bendatand Piersol (1986). The application of the HT to a signal provides some additionalinformation about amplitude, instantaneous phase and frequency which is given inthe following.Generally, the Hilbert transform H(x(t)) of a real-valued function x(t) extended

from �1 to +1 is a real-valued function defined by

HðxðtÞÞ ¼ yðtÞ ¼ lime!0

ð0�e

�1

xðuÞpðt� uÞ duþ

ð10�e

xðuÞpðt� uÞ du

� �ð2Þ

assuming thatÐ1�1ðxðtÞÞ2dt < 1, Eq. (2) can be rewritten as

HðxðtÞÞ ¼ yðtÞ ¼ 1

pp:v:

ð1�1

xðuÞt� u

du ð3Þ

where p.v. implies that the integral is meant as its Cauchy principle value. Thus, y(t)is the HT of the initial process x(t). Braun and Feldmann (1997) interpret the HT

as a natural 90vphase shifter, since y(t) is p=2 out of phase to x(t). The power

(energy) of both signals is identical. The HT forms the basis of the definition of ananalytical signal. This is the natural extension of real-valued signals to complex-valued signals which is one of the cornerstones in the discipline of modern signalprocessing (Feldmann, 1997). Then the complex-valued analytical signal z(t) isdefined as

zðtÞ ¼ xðtÞ þ iyðtÞ ð4ÞThe advantage of this representation lies in the fact that the possibility arises of

uniquely determining genuine time-variant variables. These are the instantaneousparameters with amplitude as AðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixðtÞ2 þ yðtÞ2

p, phase hðtÞ ¼ arctanðyðtÞ=xðtÞÞ

and frequency x(t) that is defined as the rate of change of the phase h(t) of the ana-lytical signal z(t) so that xðtÞ ¼ dhðtÞ=dt and, finally, Eq. (4) can be rewritten as

zðtÞ ¼ AðtÞeihðtÞ ð5ÞThese definitions form the basis of a time-frequency representation and is,

especially, convenient for situations where the product of time duration and signalfrequency bandwidth is sufficiently large. Such situations exist, for example, fornarrow-banded signals of harmonic oscillating character where the analysis via theHT shows considerable potential. However, it is known (Braun and Feldmann,1997) that problems in the physical meaning of the IF arise when the HT is carriedout on multicomponent signals, for example, chirps (swept sines) whose frequencyvaries significantly with time. Nonetheless, the IF was first introduced by Ville


(1948), but it is still highly controversial (Boashash, 1992; Cohen et al., 1999), sincefrequency is usually related with the number of cycles undergone one time by abody in a periodic motion, so that there is an apparent paradox in associating thewords ‘‘instantaneous’’ and ‘‘frequency’’. However, in practice, signals are nottruly sinusoidal, or even aggregates of sinusoidal components, representing the IFas an excellent descriptor of several physical phenomena. Originally, this conceptwas introduced in the context of AM and FM-modulations in the theory of com-munications (Ville, 1948) where it is still in extensive use. However, those historicalbackgrounds are mostly well founded and effectively adopted in applied mathemat-ics and engineering, but we should direct our focus on the essential part inside theempirical sifting process of the EMD. That is, the linking procedure of the localmaxima and minima, together with other characteristic points from the signal by acubic spline interpolation, forms the upper and lower envelopes of the signal. Littleis known about proper generation routines in this context; hence, the following sec-tion explains the cubic spline technique in depth as well as some technical improve-ments in its application for the EMD.

4. Spline interpolation

In general, the goal of a spline interpolation is to create a function whichachieves the best possible approximation to a given data set. For a smooth andefficient approximation, one has to choose piecewise higher order polynomialapproximation. A popular and practically the most applied choice is the piecewisecubic approximation function (de Boor, 1978) of order three.

4.1. Mathematical background

In comparison to interpolation polynomials, the cubic spline has minimum oscil-latory behaviour that results in smooth transitions between the data points—mak-ing the curve visually satisfying. Mathematically, a cubic spline is a third orderpolynomial applied to subsets of user defined pairs of xi and yi knots. The origin ofcubic spline functions is practically based on constructions in which a flexible ruleris used to cover a certain number of given nodal points (see Fig. 1). Under specificrequirements concerning the elasticity of the ruler, the developed curve can beapproximated piecewise by polynomials of the third degree that the first two deri-vatives are continuous everywhere. In that case, it is a cubic spline.A set of nþ 1 points (x0, y0),. . ., (xn, yn), where n � 3 (Weisstein, 2001) can be

linked by a third order spline polynomial in one-dimensional form that is generallydefined by three conditions:

1. p(x) in each interval [xi; xiþ1], i ¼ 0; 1; . . . ; n� 1 is given by a polynomial pi(x) ofa third order degree:

piðxÞ ¼ ai þ biðx� xiÞ þ ciðx� xiÞ2 þ diðx� xiÞ3 ð6Þ


2. In each interval [xi; xiþ1] i ¼ 0; 1; . . . ; n� 1, the polynomial pi(x) complies withthe following two boundary conditions:

piðxiÞ ¼ yi and piðxiþ1Þ ¼ yiþ1 for i ¼ 0; 1; . . . ; n� 1 ð7Þ3. With the restriction that each polynomial is everywhere in [x0, xn] and continu-

ously differentiable twice, two neighbored polynomials pi�1ðxÞ and pi(x) fulfillthe following two compatibility conditions:

p0i�1ðxiÞ ¼ p0iðxiÞ for i ¼ 1; 2; . . . ; n� 1p00i�1ðxiÞ ¼ p00i ðxiÞ for i ¼ 1; 2; . . . ; n� 1

ð8Þ

Consequently, there are 4�n unknown parameters (a0 . . . an�1, b0 . . . bn�1,c0 . . . cn�1, d0 . . . dn�1) and 4n� 4 equations:

piðxiÞ ¼ yi for i ¼ 0; 1; . . . ; n� 1piðxiþ1Þ ¼ yiþ1 for i ¼ 0; 1; . . . ; n� 1p0i�1ðxiÞ ¼ p0iðxiÞ for i ¼ 1; 2; . . . ; n� 1p00i�1ðxiÞ ¼ p00i ðxiÞ for i ¼ 1; 2; . . . ; n� 1

ð9Þ

This implies four additional equations at x0 and xn for the functions p00ðx0Þ, p000ðx0Þ,

p0nðxnÞ and p00nðxnÞ which are needed to specify the polynomial functions p0(x) andpn(x). As will be shown later, these boundary functions significantly influence thenumerical sifting process of the EMD. These conditions involve functions andderivative values at the end knots. For example, the function values must be equalat the interior knots and the first and last functions must pass through the end-points. For smoothness between each interval, the first derivatives at the interiorknots must also be equal. With constraints on the second derivatives at theendpoints, the boundary conditions also classify the type of the cubic spline.Concerning the EMD, the spline fitting to define the upper and the lower envel-

opes of a function rather seems to be an easy task as the only condition which is

Fig. 1. Spline interpolation.


taken into account is the proper fitting of the splines from the local extrema of the

data series. On the practical side, serious problems can occur near both ends of the

time series, where the spline fitting is prone to have large swings. Left to them-

selves, these end swings can eventually propagate inward and corrupt the whole

data series and, consequently, eliminate the natural embedded characteristic of the

data. In this case, the EMD gets totally disrupted. Additionally, overshoots and

undershoots can occur near the end, denoting the need to define specific end

conditions, as well as predefined characteristic points of the function, generally

applicable to every function, to fix the spline to the function.A trivial solution to this difficulty is to ramp the data series by ordinary window-

ing functions, e.g. Kaiser, Henning, Hamming (Bendat and Piersol, 1986), at both

ends so that the boundary effects are effectively diminished. However, on the other

side, this rather harsh technique destroys useful embedded information of the data

simultaneously. That those effects could be diminished in alternative, more elegant

ways is part of the following subsections.Figs. 2 and 3 take this statement into account. The dissimilarity between two dif-

ferent kind of end conditions is shown, whereas the spline in Fig. 2 is left to itself

without prescribing any specific determination of the end conditions. The spline

in Fig. 3 is controlled by the mathematical regulation that the second derivatives

p000ðx0Þ and p00n�1ðxnÞ at the first point x0 and the last point xn are matched by

constant values y000 and y00n , respectively, according to end condition (4) presented in

the following enumeration.To illustrate some of the possible cubic spline end conditions here, a short

description of these regulations which permits specification of many different kinds

of end conditions is given. According to de Boor (1998), a cubic spline is con-

structed with an additional set of end conditions at the first and at the last data

Fig. 2. Overshoot at the end.


sites. The following enumeration briefly reveals several usual end conditions to be

used in the context of cubic spline interpolations.

1. First derivatives p000ðx0Þ and p00n�1ðxnÞ at the first point x0 and the last point

xn are matched by constant values y00 and y0n, respectively (‘‘complete’’),2. third derivatives p0000 ðx1Þ and p000n�1ðxn�1Þ are continuous at the second x1 and

second last point xn�1 so that these points become inactive knots (‘‘not-a-knot’’),

3. first p00ðx0Þ and second derivatives p000ðx0Þ at the first point x0 are matched with

those p0n�1ðxnÞ and p00n�1ðxnÞ at the last point xn (‘‘periodic’’),4. second derivatives p000ðx0Þ and p00n�1ðxnÞ at the first point x0 and the last point

xn are matched by constant values y000 and y00n, respectively (‘‘second’’),5. second derivatives p000ðx0Þ and p00n�1ðxnÞ at the first point x0 and the last point

xn are equal to zero, respectively (‘‘variational’’).

The latter condition (5) represents a so-called natural cubic spline. By evaluating

these kinds of different end conditions to usual wave data series within this investi-

gation, the cubic spline interpolation using conditions (4) and (5) revealed to be the

most adequate, at least concerning our field of investigation. Whether the other lis-

ted conditions could be even more effective for other natural data sets—apart from

usual wave data—was not proven in the present study. Instead, the aspect of gen-

erating artificial data points produced through the data series itself turned out to

be most functional. With the support of these additional, fixed points, the irregu-

larity of the spline between two successive extrema, especially at the front and the

rear of a signal, can be minimized. Nonetheless, unwanted swings still lead to cer-

tain inaccuracies in the approximation of the cubic spline generation routines as

the EMD is yet a numerical procedure solely based on an empirical algorithm.

Fig. 3. Smooth spline fitting.


These effects occur more or less regularly. It is hardly possible to avoid these inac-

curacies on the whole, although these contradictions could be minimized. The fol-

lowing sections attempt to find alternative solutions.

4.1.1. Additional boundary data pointsIt has been already illustrated that the cubic spline has somehow to be kept to

the function especially near both ends of the data series. Therefore, the creation of

additional boundary data points, which are supposed to be applicable to the cur-

rent data set, appears to be the key element in technically improving the EMD. All

artificially added boundary data points are generated from within the original set

of discrete knots to represent a characteristic natural behaviour.One routine is to add new maxima and minima to the front and rear of the data

series. As a basic requirement, these data points are located off the original time

span the signal was recorded in. Therefore, no information is cancelled out and the

natural data series remains unaffected. In principle, Fig. 4 graphically presents this

routine for the front and Fig. 5 repeats this technique for the rear of an arbitrary

signal. This procedure is particularly implemented in the general EMD algorithm

after the extrema of the signal have been exposed. Then, for the front of the signal,

the time gaps Dtmax;f ¼ max1 �max0 and Dtmin;f ¼ min1 �min0 between the first

two successive maxima and minima are determined. The new boundary extrema

max�1 and min�1 are then shifted according to the corresponding time gaps Dtmax,fand Dtmin,f from max0 and min0, respectively. This procedure has to be similarly

repeated for the generation of additional boundary knots at the end of the data

series (Fig. 5).

Fig. 4. Additional boundary data points (front).


Subsequently, the abscissae coordinates of the new extrema are located at:

max�1 ¼ max0 � Dtmax;fmin�1 ¼ min0 � Dtmin;f

�at the front

maxnþ1 ¼ maxn þ Dtmax;rminnþ1 ¼ minn þ Dtmin;r

�at the back

ð10Þ

while the ordinate coordinates of these new boundary knots are positioned at

gðmax�1Þ ¼ gðmax0Þgðmin�1Þ ¼ gðmin0Þ

�at the front

gðmaxnþ1Þ ¼ gðmaxnÞgðminnþ1Þ ¼ gðminnÞ

�at the back

ð11Þ

Finally, to derive the new polynomials p�1(t) and pn(t) for the upper and lowerenvelopes which are supposed to represent constant functions (see Figs. 4 and 5) inthe newly created intervals [x�1, x0] and [xn; xnþ1], the appropriate end conditionsare formulated through a combination of specifications (1) and (4) from the abovelisted enumeration, and are thus easily defined as:

p0�1ðtÞ ¼ p00�1ðtÞ ¼ p0nðtÞ ¼ p00nðtÞ ¼ 0 ð12Þ

to get the cubic spline approximation of the upper and lower envelopes of theextrema of the original data series. A more sophisticated method of fix the cubicsplines at both the boundaries of the data series is principally shown in Figs. 6 and 7.By this extended method, new maxima and minima of the data series are gener-

ated using two mathematically defined slopes created through the data itself. Thesegradients represent a kind of natural steepness derived from amplitude differencesand distances between successive minima and maxima. Fig. 6 considers the casewhen a maximum is the first extremum (max0 < min0) in the front of a signal.

Fig. 5. Additional boundary data points (rear).


Then slope1,f is defined as

slope1;f ¼gðmax1Þ � gðmin0Þ

max1 �min0¼ gðmin1Þ � gðmax0Þ

min1 �max0

ð13Þ

and correspondingly slope2,f is given by

slope2;f ¼gðmin0Þ � gðmax0Þ

min0 �max0¼ gðmax0Þ � gðmin0Þ

max0 �min0

ð14Þ

Fig. 6. Additional boundary data points (front).

Fig. 7. Additional boundary data points (rear).


Formulae for both slopes in brackets in Eqs. (13) and (14) have to be taken intoaccount when a maximum is the second extremum (max0 > min0) in the front ofthe signal.Then for both cases, time gaps Dtmax;f ¼ max1 �max0 and Dtmin;f ¼ min1 �min0

between the first two successive maxima and minima are determined. The newboundary extrema max�1 and min�1 are newly defined and shifted according to thecorresponding time gaps Dtmax,f and Dtmin,f, and gradients slope1,f and slope2,f frommax0 and min0, respectively.Subsequently, the abscissae coordinates of the new extrema are located at:

max�1 ¼ max0 � Dtmax;fmin�1 ¼ min0 � Dtmin;f

�at the front ð15Þ


gðmin�1Þ ¼ ðslope1;f þ slope2;f ÞDtmin;f þ gðmin0Þgðmax�1Þ ¼ ðslope1;f þ slope2;f ÞDtmax;f þ gðmax0Þ


This procedure has to be similarly repeated for the generation of additionalboundary knots at the end of the data series (Fig. 7) which considers the first casewhen a minimum is the last extremum (maxn < minn). After having determined thetime gaps Dtmax;r ¼ maxn �maxn�1 and Dtmin;r ¼ minn �minn�1 between the last

two successive maxima and minima at the rear of the signal, the correspondingslope1,r is defined as

slope1;r ¼gðmaxnÞ � gðminn�1Þ

maxn �minn�1¼ gðminnÞ � gðmaxn�1Þ

minn �maxn�1

ð17Þ

and accordingly slope2,r is given by

slope2;r ¼gðminnÞ � gðmaxnÞ

minn �maxn¼ gðmaxnÞ � gðminnÞ

maxn �minn

ð18Þ

Formulae for both slopes in brackets in Eqs. (17) and (18) have to be taken intoaccount when a maximum is the second extremum (maxn > minn) on the rear sideof the signal. Then for both cases, the new boundary extrema maxnþ1 and minnþ1are newly defined and shifted according to the corresponding time gaps Dtmax,r andDtmin,r, and gradients slope1,r and slope2,r from maxn and minn, respectively.Subsequently, the abscissae coordinates of the new extrema are located at:

maxnþ1 ¼ maxn � Dtmax;rminnþ1 ¼ minn � Dtmin;r

�at the back ð19Þ


gðminnþ1Þ ¼ ðslope1;r þ slope2;rÞDtmin;r þ gðminnÞgðmaxnþ1Þ ¼ ðslope1;r þ slope2þrÞDtmax;r þ gðmaxnÞ


Finally, to derive the new polynomials p�1(t) and pn(t) for the upper and lowerenvelopes which are supposed to represent natural functions according to the


natural data series (see Figs. 6 and 7) in the newly created intervals [x�1, x0] and[xn; xnþ1], the appropriate conditions are formulated through a combination of spe-cifications (1) and (4) from the above listed enumeration, and are thus relativelyeasy defined. The second derivatives of the polynomials p�1(t) and pn(t) are given by

p00upper;�1ðtÞ ¼gðmax0Þ � gðmax�1Þ

max0 �max�1

p00lower;�1ðtÞ ¼gðmin0Þ � gðmin�1Þ

min0 �min�1

9>>=>>; at the front ð21Þ

p00upper;nðtÞ ¼gðmaxnþ1Þ � gðmaxnÞ

maxnþ1 �maxn

p00lower;nðtÞ ¼gðminnþ1Þ � gðminnÞ

minnþ1 �minn

9>>=>>; at the back ð22Þ

while the first derivatives of the polynomials p�1(t) and pn(t) are consequentlydefined through:

p0upper;�1ðtÞ ¼ �slope2;fp0lower;�1ðtÞ ¼ slope2;f


p0upper;nðtÞ ¼ slope1;rp0lower;nðtÞ ¼ �slope2;r


to get the cubic spline approximation of the upper and lower envelopes of theextrema of the original data series.The present method turned out to work quite successfully on wave data signals

and can be regarded as a technical improvement of the numerical sifting process ofthe EMD. Its use is heavily recommended for any analysis undertaken with theHHT. Nonetheless, we certainly do not claim that the latter approach is the ulti-mate solution to fix the cubic spline interpolation within the sifting process of theEMD. It is just an attempt that seems to work quite successfully.

4.1.2. Additional interior data pointsTo avoid further irregularities of the spline approximation, it may also be useful

to link the spline to additional characteristic points inside the signal itself. Theseare generally applicable to every natural data selection. Apart from detecting andlinking minima and maxima of a function, additional interior knots lay their focuson extended natural characteristics of the signal. These are examined according tothe following list:

. Interior inflection (or turning) points,

. Interior extrinsic curvature extrema points,

. Interior characteristic points which could be derived following the minimumvalue theorem.

As shown already, it is the fundamental process of the EMD to define the upperand lower envelopes within the numerical procedure. Huang et al. (1998, 1999)


propose that it is therefore necessary to detect all embedded extrema of the signal,

and subsequently, link these knots by cubic splines. Additionally, they suggest

determining all zero-crossings of the signal to enable a correlation of the amount

of those with the total number of extrema in order to establish stopping criteria

within the EMD. Identifying extrema and zero-crossings of a given data series x(t)

can be done analytically according to the following regulations:

x0ðtÞ ¼ 0 and xðtÞ ¼ 0 ð25Þ

where x0(t) represents the first derivative of the signal x(t) in a closed analytical

form. Experimental investigations urge defining the continuous data series x(t) at

discrete time steps ti to get a discrete representation at xi with i ¼ 0; 1; . . . ; n

(n being the total number of discrete knots), simply due to the fact that the finite

sampling interval Dt is always given in practical applications, so that time steps get

ti ¼ iDt. Accordingly, total sampling duration is defined as Tend ¼ Dtn. Finally,Eq. (25) in differential form gets for the finding process of all zero-crossings of the

signal (terms in brackets in Eq. (25) represent types of zero-crossings):

signðxiÞ < signðxiþ1Þ ðupcrossingÞsignðxiÞ > signðxiþ1Þ ðdowncrossingÞsignðxiÞ ¼ signðxiþ1Þ ¼ 0 ðstraddlingÞ

ð26Þ

and correspondingly for the determination of the extrema with the signals’ first

derivative x0i ¼ ðxiþ1 � xiÞ=Dt, where i ¼ 0; 1; . . . ; n� 1, and second derivative

x00i ¼ ðx0iþ1 � x0iÞ=Dt, where i ¼ 0; 1; . . . ; n� 2, the maxima and minima, respect-

ively, can be calculated in differential form according to the process:

sign x0i� �

6¼ sign x0iþ1

� �� K sign x00i

� �> 0

� �K sign x00iþ1

� �> 0

� �sign x0i

� �6¼ sign x0iþ1

� �� K sign x00i

� �< 0

� �K sign x00iþ1

� �< 0

� � ð27Þ

Additional interior knots first take the inflection (or turning) points under further

consideration. These characteristic points are significant marks for natural changes

in the curvature behaviour of the signal and are defined as the inverse of the radius

of the curves’ osculating circle. In general, the inflection points are easily detected,

when obeying the following rules:

sign x00i� �

< sign x00iþ1

� �� ðright left inflectionÞ

sign x00i� �

> sign x00iþ1

� �� ðleft right inflectionÞ ð28Þ

Calculating characteristic knots from Eq. (28) notably supports defining the

upper and lower envelopes within the EMD. The procedure can be taken to a

higher stage when deriving curvature extrema of the signal. The simplest form of

curvature in one dimension is an extrinsic curvature j(t). It classifies the reciprocalvalue of the radius R of an arbitrary curve osculating circle and has units of

inverse length. Usually, it is encountered in calculus and is defined in closed


analytical form of a signal x(t) by

jðtÞ ¼d2xðtÞdt2

1þ dxðtÞdt

2 !3=2

ð29Þ

and can subsequently be transformed into a discrete representation in the generalform: ji ¼ x00i =ð1þ ðx0iÞ2Þ3=2, with i ¼ 0; 1; . . . ; n� 2. The latter equation is set equalto zero in order to determine curvature extrema points of the signal.The predominant task is now to numerically define extrema in the changes of the

curvature ji. This can be done by

signd

dtji

6¼ sign

d

dtjiþ1

K sign

d2

dt2ji

>0

K sign

d2

dt2jiþ1

>0

� sign

d

dtji

6¼ sign

d

dtjiþ1

K sign

d2

dt2ji

<0

K sign

d2

dt2jiþ1

<0

ð30Þ

It is found that retrieving distinct information about the curvature extrema of thesignal can foster an appropriate definition of the upper and lower envelopes withinthe EMD. Again, this attempt should not be regarded as the ultimate stage in solv-ing the cubic spline question.Potentially, a very sophisticated method has been introduced by Qiang et al.

(2001) in using the mean value theorem for detecting further characteristic points inthe data series. They suggest determining interior points for creating the upper andlower envelopes according to the principle that for any differentiable function x(t)on the open interval (a, b) and continuous on the closed interval [a, b], at least onepoint c in (a, b) exists such that

x0ðcÞ ¼ xðbÞ � xðaÞb� a

ð31Þ

Concerning our job within the numerical process of the EMD, we can simply detectcharacteristic data by calculating the local linear gradient between two successiveextrema. Then, the theorem states that there is at least one point inside this specificinterval of the data series that exposes an equal gradient. This routine is supposedto work quite successfully, as Qiang et al. (2001) point out that it explicitly uses themean value theorem to determine the local mean without the numerical sifting pro-cess of the EMD. They conclude that this algorithm is quicker than the conven-tional one and extend their method by finding artificially created data points at thefront and at the end, beyond the limits of the data series by using a two-tap adapt-ive time-varying filter. Further particulars on this subject can be found in Qianget al. (2001). However, within the present investigation, it was found by the authorsthat this promising approach provides no further technical improvement of theEMD. Subsequently, it has been discarded from further studies.


5. Investigation of parameter-interference

In the following section, the results of a simple parameter study on the perform-ance of the EMD are presented to reveal a better understanding of this numericalmethod and to illustrate its restrictions and limitations. On the whole, this empiri-cal parameter study turns out to be straightforward in learning how the EMDnumerically performs on trigonometric functions, and consequently, derives prob-able answers with a certain degree of accuracy on what we can expect to emergefrom this time-frequency method for any arbitrary data series.

5.1. Variation of frequency and amplitude proportions of time-invariant components

It seems clear that within the decomposition process of a given data series, anumber of arbitrary chosen parameters are prone to affect the overall success ofthe analyzing procedure. First, we attempt to enumerate those and investigatewhich are the most significant and how do these parameters influence each other toformulate precisely what kind of constellation corollaries in an overall inaccuracyof the decomposition process. Here, we regard the simple example of two super-imposed sine waves. In the first step, both components coincide in phase buildingoverall a stationary process since amplitudes and frequencies are time-invariant.The essential parameter configurations which expose having a dominant influenceon harmonic data series are elucidated. To achieve these objectives, the parametersof the trigonometric functions are varied systematically regarding their individualeffect on the accuracy of the EMD in separating the origin of analytically exactdefined components. It seems clear that Fourier analysis based techniques wouldprincipally perform as well or even better as the EMD since linear and time-invariant functions are treated in this very first step. However, this statement onthe performance of Fourier methods turns invalid for time-varying frequencies andamplitudes of specific components. But the EMD will reveal its full physicalinsightful power on these configurations as will be shown later. In the first stage,we deal with stationary data series—one of the trigonometric functions x1(t) is sup-posed to be the carrier wave. It depends on amplitude a1 and frequency f1. Anothertrigonometric function x2(t) is taken and consequently named riding wave. It isgoverned by amplitude a2 and frequency f2. It was figured out to have at least afrequency relation f 2=f 1 � 1:1 since the EMD is categorically incapable of numeri-cally disintegrating components that have frequency proportions near unity. Thisturned out to be one of the first mathematical limitations of the EMD—thisnumerical procedure is categorically unable to distinguish between harmonic func-tions that have small frequency relations, lesser than 0.1 Hz. Both sine waves aresuperimposed to artificially generate a kind of bichromatic data series X(t) withtime-invariant components:

XðtÞ ¼ a1sinðf1tÞ þ a2sinðf2tÞ with 0 � t � Tend ð32ÞNote that here X(t) is given in a closed analytical form. To proceed with the

present investigation and evaluate the performance of the EMD, the data serieshas to be digitized. Consequently, the continuous data series X(t) is projected at


distinct time steps of ti in order to get a discrete representation of the super-imposed sine waves at Xi, with i ¼ 0; 1; . . . ;N (N being the total number of knots).This parameter study varies the total number of knots N as well as the total sam-pling duration defined as Tend. Accordingly, the finite sampling interval is definedas Dt ¼ Tend=N. In practical applications, time steps then become ti ¼ iDt so thatEq. (32) in discrete notation turns out as,

xi ¼ a1sinðf1tiÞ þ a2sinðf2tiÞ with ti ¼ iDt and i ¼ 0; 1; . . . ;N ð33ÞKnowing the superimposed input Xi of these two trigonometric functions x1 and

x2, it is obvious how both resulting IMFs as the key output of the EMD have tobe calculated precisely. It is then easy to derive numerical quantities in between theknown input and computed output, and, therefore, give details of the numericalperformance of the EMD. In this context, we derive both the root mean square(rms) and the ‘‘linear’’ correlation coefficient (corrcoef) between the input and out-put data. While the first examines the absolute inaccuracy of the EMD in analogyof a basic standard deviation, the latter determines a relative assessment for theembedded error under the crucial assumption to deal with linear processes. Thecorrelation coefficients are presented hereafter since it was found that these inde-pendently determined quantities show identical tendencies and behave analogously.Accordingly, the root mean square values, rms1 for the carrier, and rms2 for theriding wave, are calculated using (Bronstein, 1999)

rmsh1;2i ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N

XNi¼1

xih1;2i � IMFih2;1i

� �2vuut ð34Þ

In correspondence, both correlation coefficients corrcoef1 and corrcoef2 for thecarrier and riding waves, respectively, are evaluated by means of (Bronstein, 1999)of

corrcoef h1;2i ¼covðxh1;2i; IMFh2;1iÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

varðxh1;2i; xh1;2iÞvarðIMFh2;1i; IMFh2;1iÞp ð35Þ

Note that in Eq. (35) ‘‘cov’’ represents the covariance and ‘‘var’’ the variance ofthe corresponding data. Table 1 provides an in-depth view on these systematicvarying parameters. Proportions of amplitudes a1=a2 and frequencies f 2=f 1 are

Table 1

Parameter study

Parameter
Symbol R ange
Amplitude proportion
a1=a2 [ 1.0, 5.0, 10.0]
Frequency proportion
f 2=f 1 [ 1.0, 1.1, . . ., 14.9, 15.0]
Sampling duration
Tend [ 4p, 10p, 30p s]
Data points
N [ �101, . . ., 1:0009� 105]
Sampling rate
dt ¼ Tend=N [ �101, . . ., 1:25� 105 s]
Sampling frequency
F ¼ 1=dt [ �10�1, . . ., 8:0� 103 Hz]


systematically changed. Moreover, time support Tend and the number of total datapoints N of the data series are altered to establish a kind of empirical behaviour

study of the EMD containing more than 100,000 different experimental configura-tions. In this context, amplitude a1 and frequency f1 of the carrier wave were keptconstant and set equal to 1—consequently, a2 and f2 of the riding wave are alter-nated.In the following, the numerical performance of the EMD on the superimposed

amplitude- and frequency-altered sine waves is directly investigated within a rangeof around 101–105 data points N and indirectly with sampling frequencies F from

0.1 up to approximately 8000 Hz. Computing time for all listed configurations con-sumes about 8� 105 s on a Pentium III (850 MHz) running Matlab 6.1 (R12)—just to document this extensive empirical analysis. For example, Figs. 8 and 9present calculated results of correlation coefficients for both the riding and carrierwaves, respectively, for an amplitude proportion a1=a2 ¼ 1 and variable time sup-

port Tend ¼ 4p, 10p and 30p. The coefficients are shown over frequency propor-tions f 2=f 1 versus sampling frequency F in logarithmic scale. In addition, Figs. 10and 11 show the root mean squares between input and output data under identicalfrequency proportions and time support previously used in Figs. 8 and 9.Apparently, no substantial differences or significant discrepancies in between rms

and corrcoef magnitudes are detectable. Other experimental configurations alsoreveal this remarkable equivalence—it was therefore concluded to proceed onlywith the correlation coefficient representations in the following for the sake of

brevity. It is evident from Figs. 8–11 that the EMD disintegrates both trigono-metric components with amplitudes a1 ¼ a2 ¼ 1 very accurately for frequency pro-

portions f 2=f 1 > 2 and higher sampling frequencies F > 101 Hz. Thus, these areas

n coefficient R2 between riding wave x2 and IMF1 (a1=a2 ¼ 1, Tend ¼ 4
Fig. 8. Correlatio p, 10p and 30p).


are indicated by a large magnitude of the correlation coefficients R2 > 0:90,although sampling frequencies F play an insignificant role in distinguishingbetween riding and carrier waves for relatively low frequency relations. Correlationcoefficients and root mean square magnitudes also show noteworthy mathematical

deviations for low sampling frequencies F < 101 Hz and large frequencyproportions. The technique is incapable of deriving accurate results under these

ion coefficient R2 between carrier wave x1 and IMF2 (a1=a2 ¼ 1,
Fig. 9. Correlat Tend ¼ 4p, 10pand 30p).
an square (rms) between riding wave x2 and IMF1 (a1=a2 ¼ 1, Tend ¼ 4p
Fig. 10. Root me , 10p and 30p).


mathematical boundaries obviously. Calculations become more precise (R2 > 0:90)in relation to an increasing frequency relation and higher sampling frequenciesF. Dashed black lines in all contour plots represent the Nyquist frequencies withregard to ordinary Fourier analysis techniques. Most interestingly, the EMD math-ematically behaves in a similar manner. Shannon’s sampling theorem (Bendat andPiersol, 1986) states that within an arbitrary data series, any embedded highfrequency component with frequency M has to be sampled at least with double the

frequency 2M to get reconstructed from the signal. The EMD that constitutes acompletely different mathematical background reveals identical physical quantities,although it is more capable of deriving components that show larger frequenciesmore precisely than the Nyquist frequencies. In a Fourier world, these outliners arebackfolded into the spectra—a process that is better known as ‘‘Aliasing’’—andconsequently determines incorrect spectral representations of data series. However,it is out of scope of the present paper to determine those underlying mathematicalmechanisms to clarify why and how the EMD either performs well or mismatchesthe disintegration of the input signal for components with frequencies violatingShannon’s sampling theorem. The aim is more to document the behaviour of theseparation process within the EMD.Figs. 12 and 13 present results for an amplitude proportion between the carrier

and riding waves of a1=a2 ¼ 5. All other parameters are kept constant with regardto the preceding configuration. Obviously, calculating the linear correlation coeffi-cients reveals a less effective performance of the EMD with regard to the latter

experimental arrangement. Here, accurate results (R2 > 0:90) are gained for

frequency proportions f 2=f 1 > 3 and higher sampling frequencies F > 101 Hz.

an square (rms) between carrier wave x1 and IMF2 (a1=a2 ¼ 1, Tend ¼ 4
Fig. 11. Root me p, 10p and 30p).


Correlation coefficient magnitudes again indicate cumbersome mathematical

deviations for low sampling frequencies F < 101 Hz and large frequency propor-

tions—the EMD characterizes the lack of ability to derive precise results under

these mathematical boundaries. Dashed lines in both contour plots again represent

the Nyquist frequencies. As for the latter configuration, it is obvious that EMD

on coefficient R2 between riding wave x2 and IMF1 (a1=a2 ¼ 5, Tend ¼ 4
Fig. 12. Correlati p, 10p and 30p).
on coefficient R2 between carrier wave x1 and IMF2 (a1=a2 ¼ 5, Tend ¼ 4


mathematically behaves in a similar manner than the natural limitation from a

Fourier-based analysis. In a further step, Figs. 14 and 15 show results for a larger

amplitude proportion a1=a2 ¼ 10 between carrier and riding waves. Astonishingly,

contour plots reveal that the EMD is even less effective for this experimental

configuration than those carried out earlier. Here, accurate results (R2 > 0:90) are

on coefficient R2 between riding wave x2 and IMF1 (a1=a2 ¼ 10, Tend ¼ 4
on coefficient R2 between carrier wave x1 and IMF2 (a1=a2 ¼ 10, Tend ¼ 4


achieved just for frequency proportions f 2=f 1 > 5 and higher sampling frequencies

F > 101 Hz. Below these boundaries, it is incapable of disintegrating the super-imposed bichromatic function into its individual time-invariant components.Hence, some empirical performance rules of the EMD according to the definitionof the linear correlation coefficient (and root mean squares) for the two super-imposed trigonometric functions are derived. Table 2 gives a brief summary of theexpected correlation coefficients for both riding and carrier waves being at least

R2 > 0:9 according to a given amplitude proportion a2=a1. The polynomial fittedfunction shown in Table 2 is extracted from results of all three experimental config-urations. It is thus being characterized as a lower numerical disintegration bound-ary of the EMD and the polynomial fitted function yields the minimum frequencyrelation needed to resolve carrier and riding waves with the given amplitudeproportion a1=a2. Shannon’s sampling theorem consequently forms the upper-left boundary for the useful application of this newly developed time-frequencyrepresentation. Ergo, trigonometric superimposed functions violating these rules,the EMD—as well as Fourier-based techniques—are mostly incapable of distin-guishing between carrier and riding components.On the whole, the more the frequency information between carrier and riding is

equal, the worse the disintegration of both trigonometric waves. Moreover, thecoarser the riding and carrier waves are mathematically resolved through the sam-pling frequency F, the worse the correlation coefficient. This is found to be in ana-logy with Shannon’s sampling theorem which states that any Fourier-basedanalysis technique is only capable of determining components with half the sam-pling frequency the data series that has been recorded. Concerning this particularFourier limitation, the EMD is an even more adequate tool to determine high fre-quency components within signals without sufficient small sampling rates. Further-more, it is now known that it is also useful to add more information to the dataseries and, therefore, improve the embedded internal contents through upsampling(or oversampling) techniques based on spline interpolation routines. Due to theseartificial data enrichment measures, the analysis performance and mathematicalaccuracy of the EMD significantly increase as the present investigation based onparameter study shows.

5.2. Phase-shifted time-invariant components

So far, basic investigations have been performed under the crucial assumption ofdealing with trigonometric components coinciding in phase. However, examining a

Table 2

Amplitude- and frequency-proportions

a1=a2 ¼ 1 a
1=a2 ¼ 5 a1=a2 ¼ 10 a1=a2 ¼ x
f 2=f 1
2 3 5 ¼ 160x
2 þ 320 xþ 11

6


broader frame of special constellations becomes necessary since real-world signalsare unlikely to be recorded when both or all components—dominating the process—coincide in phase. Hence, a very brief analysis on the performance of the EMD iscarried out when dealing with the particular cases when carrier and riding wavesexperience a certain degree of phase lag h defined in between 0 � h � 2p. Its influ-ence on the accuracy of the EMD is investigated systematically in another typicalparameter study. Figs. 16–19 give a brief overview on how the EMD separates

coefficient R2 between riding wave x2 and IMF1 with altered ph
Fig. 16. Correlation ase lag h on riding
wave (f 2=f 1 ¼ 2, a1=a2 ¼ 1, F ¼ 10 Hz, Tend ¼ 4p, 10p and 30p).



riding and carrier waves with phase variation. At the same time, it also reveals the

complexity of this analysis in interpreting results. For example, Figs. 16 and 17

present correlation coefficients R2 for input and output riding and carrier waves

(f 2=f 1 ¼ 2, a1=a2 ¼ 1, F ¼ 10 Hz), respectively, with a systematically changed phase

h. The phase lag is previously added onto the riding wave of the input signal.Results from the latter configuration on R2 without phase deviations are found

for h ¼ 0 and can be correspondingly obtained from Figs. 8 and 9. Interestingly,

Fig. 18. Correlation ase lag h on carrier

coefficient R2 between carrier wave x1 and IMF2 with altered ph


correlation coefficients show significant deviations over the varied phases. Fig. 16

shows that the riding wave can be reproduced from the input signal very accurately

(R2 > 0:9) more or less without any deviations depending on the total sampling

duration. It is noteworthy that two maxima of the correlation coefficient around

p=2 and 3p=2 are visible and point to the fact that the EMD performs very effec-

tively when both components are orthogonal in the mathematical sense. It has

been additionally observed that this separation effect becomes even more dominant

for higher sampling frequencies F under identical configuration for the riding wave.

The correlation coefficients derived from disintegrating input and output carrier

wave in Fig. 17 show another tendency. While the EMD mainly separates the car-

rier wave without a significant influence of the phase variation for large sampling

durations, a considerable dependency on the phase is evident for short sampling

duration. This outcome was previously also embedded in Fig. 9, but did not

appear visible at first sight due to graphical smoothing filters of the three-dimen-

sional contour plot. However, dependencies on the total sampling duration are

obvious, and it is disclosed that the EMD is incapable of determining a correlation

between input and output data when the phase variation between riding and car-

rier waves is approximately 3p=4 � h � 11p=8. But again, it is out of scope of thepresent paper to clarify why this technique mismatches the disintegration of the

input signal for components which are out of phase—it is more a kind of a general

description on how the EMD operates on phase-shifted trigonometric functions.

Nevertheless, this troublesome feature has to be studied in more detail in future

investigations on the EMD.Figs. 18 and 19 present correlation coefficients R2 for input and output riding

and carrier waves, respectively, with a systematically changed phase h, where the

phase lag was now added onto the carrier wave of the input signal. At a first

glance, results are dissimilar compared to those in Figs. 16 and 17. Fig. 18 shows

that the riding wave can again be reproduced from the input signal quite accurately

(R2 > 0:9) almost without any deviations depending on the total sampling dur-

ation—except for the shorter time periods when the signal is recorded. Now, four

cumbersome maxima of the correlation coefficient around p=4, 3p=4, 5p=4 and 7p=4are visible and point to the fact that the EMD performs very effectively on the

riding wave when both input components are phase shifted in these particular pro-

portions. This is in clear correspondence to the two maxima around p=2 and 3p=2from Fig. 16 where the phase lag has been added onto the input riding wave.

Again, it is stunning that the EMD disintegrates both components very accurately

when they are orthogonal to each other. The correlation coefficients derived from

disintegrating input and output carrier wave in Fig. 19 show an identical tendency

previously derived from Fig. 17. Again, a considerable dependency on the phase is

evident for short sampling duration. It is therefore determined that the EMD is

incapable of determining a correlation between input and output data when the

phase variation between riding and carrier waves is approximately p=8 � h �15p=8 for short sampling duration. As a rule of thumb, it can be concluded that


the longer the time period of the record, the better the disintegration results fromphase-shifted components.For example, Figs. 20–23 show results of the correlation coefficient for another

experimental configuration. Corresponding to the latter figures, calculations ofR2 for an amplitude proportion a2=a1 ¼ 5, frequency proportion f 2=f 1 ¼ 5 andsampling frequency F ¼ 100 Hz are presented. According to Fig. 20, influence ofmagnitudes of the phase difference between riding and carrier waves is less




significant for this experiment. An almost overall constant value of the correlationcoefficient for the riding wave is revealed, although the known dependency on thetotal sampling duration is obvious. The local maxima appear at h ¼ p. The EMDdisintegrates best for this specific phase lag between riding and carrier waves.Results for calculations of the riding wave are shown in Fig. 21. An outstandingperformance is revealed—entirely independent of the total sampling duration ofthe data series. Calculations of the correlation coefficient get mostly confirmedwhen the phase lag is added on the carrier wave as Figs. 22 and 23 determine, even




though the common dependence on the total sampling duration is noticeable forthe riding wave. The carrier wave is again resolved very precisely.On the whole, a significant dependence on the phase variation between riding

and carrier waves is detected when dealing with the overall performance of theEMD with regard to the disintegration success of embedded components from aprincipally unknown data series. But to determine the kinds of quantitative predic-tion rules with analytical tools is categorically out of reach for this empirical tech-nique, although certain tendencies on the phase dependence are qualitativelyderived to use the EMD within any practical applications in engineering discipline.It is very well known to the authors of the paper that these limitations and short-comings of the EMD kept away theorists of data analysis so far. Some of them willcertainly avoid using the EMD in future as well and stick to their traditionalparadigm, but perhaps another fraction attempts to use this technique more fre-quently due to the present extensive investigation as the following section showsthe real advantages of the EMD right now.

5.3. Variations of time-variant components

An essential question that may come up at this point is: why should someoneuse an empirical technique to separate two (or more) components from a data ser-ies that is less efficient and, moreover, by far more complex to use and interpretthan conventional Fourier-based methods. This is easy to answer as the EMD andconsequently the HHT are capable of clearly differentiating between time-variantcomponents from any given data set. Whether these components change over timeharmonically or are of transient nature—e.g. impulse-like disturbances—is insig-nificant for this time-frequency technique. Huang et al. (1998, 1999) note that otherthan stationarity, Fourier spectral analysis also requires linearity. However, numer-ous natural phenomena have a stringent tendency to behave nonlinearly whenevertheir variations become finite in amplitude. Under these conditions, Fourier spec-tral analysis is of limited use—for the lack of alternatives, however, it is commonlyutilized in an uncritical way to process non-stationary and nonlinear data series,although this can lead to misleading results. On the contrary, the EMD/HHT arecharacterized through their essential features of effectively considering nonlinearand non-stationary data series. These attributes make this technique unique andoutstanding in the world of applied data analysis. In the following, we present theperformance of the EMD/HHT on time-variant amplitude- and frequency-modu-lated data series. For example, Eq. (36) gives one possible mathematical descrip-tion of an amplitude-modulated signal.

x1ðtÞ ¼ a1sinðf1tÞx2ðtÞ ¼

a2 þ Da22

sinf2 þ Df2

2t

þ a2 � Da2

2sin

f2 � Df22

t

XðtÞ ¼ x1ðtÞ þ x2ðtÞ

ð36Þ

defined within 0 ¼ t ¼ Tend. Again, a1 and f1 represent amplitude and frequency ofthe carrier wave, respectively, and are based on the other experimental configura-


tions of magnitude 1. The riding wave has its original mean amplitude a2 modu-lated in proportion to a bearing amplitude differential Da2, as well as its mean fre-quency f2 that is now modulated by a fraction Df2. To get an idea about theperformance of the EMD on amplitude-modulated signals, here, magnitudes ofparameters are arbitrarily set: a2 ¼ 0:5, Da2 ¼ 0:3 for amplitudes and f 2 ¼ 7 Hz, Df 2 ¼ 0:7 Hz for the frequency modulations of the whole data series. Fig. 24presents a graphical sketch of this signal which was generated over a total timespan Tend ¼ 30p with a sampling frequency of about F � 10 Hz.The EMD separates riding and carrier waves very accurately. Figs. 25 and 26

graphically present both disintegrated components. Linear correlation coefficientbetween input and output signal is R2 ¼ 0:9996 for the riding and R2 ¼ 0:9997 thecarrier waves. At this point, it must be emphasized that Fourier-based techniquesare principally capable of distinguishing between riding and carrier waves as well.Components defined in spectral representation include the corresponding phasespectra for the whole time span Tend. The signal can easily get reconstructedthrough an inverse Fourier transformation. But what is most important is thatFourier-based techniques cancel out time information so that amplitude-modulatedsignals—here the riding wave—are never represented through their originalcharacteristics. The riding wave component is defined in a Fourier spectra with atime-invariant, constant amplitude without taking its evolution in time into con-sideration. The magnitude of this component is the global mean amplitude of theriding wave. Modulations are totally neglected. This is due to the fact that Fouriertransform is a decomposition of complex exponentials, which are of infinite dur-ation and completely unlocalized in time, although time information is encoded inthe corresponding phases of the Fourier transform. Their interpretation is notstraightforward and a direct extraction is faced with a number of difficulties suchas phase unwrapping.

. Amplitude-modulated signal (Tend ¼ 30p, F � 10 Hz and N ¼
Fig. 24 1000).


The next example goes even beyond the present step as we superpose two com-ponents that are amplitude- and frequency-modulated. It is shown how the EMDperforms excellently in disintegrating both origin components. This example issupposed to bring out the essential benefits of the nonlinear, non-stationary time-frequency analyzing technique. The new data series is composed of one sinusoidalfrequency-modulated riding wave (f2;low ¼ 3 Hz, f2;up ¼ 10 Hz) that is of constantamplitude (a2 ¼ 1). The other component is a Gaussian wave packet (modulatedamplitude a1 ¼ 1 ranging from 0 to 1) of constant frequency (f 1 ¼ 3 Hz) acting as

1 of an amplitude-modulated signal (Tend ¼ 30p, F � 10 Hz and
Fig. 25. IMF N ¼ 1000).
2 of an amplitude-modulated signal (Tend ¼ 30p, F � 10 Hz and
Fig. 26. IMF N ¼ 1000).


the carrier wave. Both functions are superposed and jointly plotted in Fig. 27. The

signal was again generated in a total time span of Tend ¼ 30p with a sampling fre-

quency of about F � 10 Hz. Results from the decomposition process are plotted in

Figs. 28 and 29. Obviously, the EMD successfully derived both origin components

with correlation coefficients R2 ¼ 0:9985 for the riding and R2 ¼ 0:9998 for the

carrier waves.

plitude- and frequency-modulated signal (Tend ¼ 30p, F � 10 Hz and N
Fig. 27. Am ¼ 1000).
amplitude- and frequency-modulated signal (Tend ¼ 30p, F � 10
Fig. 28. IMF1 of an Hz and N ¼ 1000).


6. Application of irregular water waves

6.1. Introduction

The following examples introduce the application of the HHT to the field ofcoastal and ocean engineering and indicate the characteristics of nonlinear waterwaves from a different point of view. Numerical simulations have been conductedby running Matlab 6.1 (R12), whereas physical modelling tests have been carriedout in a laboratory wave flume (length l ¼ 25 m, width w ¼ 0:3 m, heighth ¼ 0:5 m, water depth d ¼ 0:3 m). Within the scope of the present article, onlybichromatic waves are treated, see e.g. Schlurmann (2000, 2002) for more detailedinvestigations on simple nonlinear monochromatic waves as well as on transientwavegroups, here, extreme waves recorded in the Sea of Japan. However, in thepresent investigation, two-component waves are calculated using Stokes secondorder theory for irregular waves (see Longuet-Higgins and Steward, 1960; Zhanget al., 1996). The model considers the effects of interaction between both wavecomponents of an irregular wave train up to second order of steepness and isprobably the most important interaction process relevant to the prediction of wavekinematics and short-distance evolution in the absence of wave breaking. Labora-tory experiments have been carried out employing a second order driving signalaccording to the usual wave maker theory (Schaffer, 1996) operating in piston-typemode. Various alternative experimental configurations were investigated in con-stant water depth d ¼ 0:3 m. In order to acquire detailed information about inci-dent wave records, water surface elevations were measured using six conventionalparallel-wire resistance wave gauges mounted in the flume in an irregular spatialpattern (X ¼ 3:70, 5.35, 6.42, 8.32, 9.96 and 12.48 m measured from the non-moving wave board, respectively). Each wave gauge has been carefully calibrated

amplitude- and frequency-modulated signal (Tend ¼ 30p, F � 10
Fig. 29. IMF2 of an Hz and N ¼ 1000).


to guarantee precisely resolved data. The present paper uses four selected time ser-

ies in particular to obtain certain physical insights about the nonlinear wave evol-

ution of bichromatic waves described by the HHT. Table 3 gives an overview

about the experimental configuration. Sampling frequency f ¼ 50 Hz is kept con-

stant for all experiments—only a few additional tests were run with f ¼ 100, 150

and 200 Hz to verify preceding investigations, but are not shown here.

6.2. Numerical simulations

The solution for the interaction between two regular wave trains with relatively

close frequencies was given by Longuet-Higgins and Steward (1960) using a con-

ventional perturbation expansion approach. The interaction taking place between

two wave trains that are well separated in frequency domain is significantly differ-

ent from the latter theory since the riding high-frequency component is strongly

modulated by the carrier wave. Assuming an incompressible, irrotational flow field

and constant pressure at the free surface, the surface elevation g(t) of a unidirec-

tional wave field is principally composed of both original fundamental components

as well as two nonlinear bounded superharmonic wave trains according to Zhang

et al. (1996). Additionally, two further nonlinear bounded components appear,

namely the sub- and superharmonic component which are due to the interaction

process of both wave trains. Eq. (37) names each constituent and presents the sum-

mation of all previously described components.

gðtÞ ¼ aicoshi|fflfflffl{zfflfflffl}1st fundamental component

þ ajcoshj|fflfflffl{zfflfflffl}2nd fundamental component

þ 1

4a2i kiai 3a

2i � 1

� �cos2hi|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

1st superharmonic component

þ 1

4a2j kjaj 3a2j � 1

� �cos2hj|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

2nd superharmonic component

þ Bð�Þcosðhj � hiÞ|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}Irregular subharmonic component

þ BðþÞcosðhj þ hiÞ|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}Irregular superharmonic component

ð37Þ

where ai,j denotes the amplitudes of the fundamental components, hi ;j ¼ ki ;jx�xi ;j tþ bi ;j the linear wave phases with ki,j the wavenumbers, xi,j the cyclic fre-

quency and bi,j the initial phase angle, and moreover, ai ;j ¼ cothðki ;jdÞ, where d

stands for water depth, respectively. In addition, the coefficients B(�) and B(+) for

Table 3

Amplitude- and frequency-proportions for three experimental configurations in a laboratory wave flume

d
(m) F (Hz) Tend (s) a1=a2 f 2=f 1
Test 1 0
.3 50 50 >1 2
Test 2 0
.3 50 50 >1 3
Test 3 0
.3 50 50 >1 4


the sub- and superharmonic components are defined as

Bð�Þ ¼ � aiajkj2aj

aik2ij � 1

� �ajk

2ij a2i � 1� �

þ 2aiajkijðaj � aiÞ � ai a2i � 1� ��

� ðaj � aikijÞ2 � ð1� kijÞ2� �

ð38Þ

where kij ¼ xi=xj. Note that integer subscripts i and j define the individual wave

component. Analogously, this numerical scheme can also be used to define randomsea states with a finite number of components, e.g. to generate transient waves inlaboratory wave flumes (see Schlurmann et al., 2000). The nonlinear dispersionrelation for the two interacting wave trains is derived in this study using the con-ventional perturbation expansion approach which can be defined as

xi ¼ gki tanhðkidÞ 1þ a2i Si þ a2i Qij

� �xj ¼ gkj tanhðkjdÞ 1þ a2j Sj þ a2i Qij

� � ð39Þ

with Si;j ¼ k2i;jðð9=8Þða2i;j � 1Þ2 þ a2i;jÞ representing the nonlinear effects of the

component amplitude on its own frequency and is identical to that in a Stokeswave solution. Furthermore, Q12 indicates the effect of the component’s amplitudeon the other component’s frequency and is therefore defined as

Qij ¼ � 1

2a2j½ðajbs � 1ÞSXij � ðajbr � 1ÞRXij �

xj

xik2j þ

1

2a2jðaj þ 1Þ

� ½1þ SXij þ RXij�k2j þaiaj

� ð1þ SXij þ RXijÞ �bs

2

1

aiþ 1

aj

SXij þ

br

2

1

aiþ 1

aj

RXij

� �xi

xjk2j

� 1

2

aiaj

a2i þ 1� �

þ 2ðRXij � SXijÞ� �

kikj

þ 1

2aj½ðai � 1ÞSXij � ðai þ 1ÞRXij þ 2aj�

xi

xjkikj

ð40Þ

where br ¼ ðaj � ajÞ=ð1� aiajÞ and bs ¼ ðaj � ajÞ=ð1� aiajÞ. Furthermore, the

coefficients RXij and SXij are defined as

RXij ¼ ðaiaj � 1Þ2kijð1� kijÞðaiaj þ 1Þ � k3ij a2i � 1

� �þ a2j � 1

2 k2ij a2j � 1� �

� 2kijðaiaj � 1Þ þ a2j � 1� �

SXij ¼ �ðaiaj � 1Þ2kijð1þ kijÞðaiaj � 1Þ � k3ij a2i � 1

� �þ a2j � 1

2 k2ij a2j � 1� �

� 2kijðaiaj þ 1Þ þ a2j � 1� � ð41Þ

Zhang et al. (1996) further note that when the water depth is deep enough(ai; aj ! 1), the above precisely given dispersion relation reduces to a form


identical to those derived by Longuet-Higgins and Steward (1960) for the specialcase of a deep water environment. The nonlinear dispersion relationships indicatethat the phase velocity increases not only due to its own amplitude but also to theinteracting amplitude when the interacting wave train advances in the same direc-tion. Consequently, according to the preceding formulae, numerical simulations areperformed and validated extensively versus the data measured in the laboratorywave flume. The results are presented in the following section.

6.3. Results

The following figures mainly reproduce results of this experimental investigationon bichromatic wave trains in a laboratory wave flume with a numerical descrip-tion given by the preceding theory and consequently the HHT. Note that Fig. 30and all the forthcoming figures are divided into 18 subfigures in total making theoverall informational contents quite complex. Each row represents the recordedwater surface elevations and analysis results from a different position in the wavetank and contains three individual subfigures. Each experiment presented hence-forth encloses a carrier wave which is of constant frequency of f 1 ¼ 0:5 Hz in0.3 m water depth, so that its wavelength consequently matches Lcarrier � 3:25 m.Therefore, each subfigure in the first column displays a dimensionless relationX=Lcarrier defining the position of the gauge in the tank and the wavelength ofcarrier wave derived from second order irregular wave theory. Riding wave’sfrequency is varied for each experimental configuration according to Table 3. Inthis context, Figs. 30–32 present results from test 1. This first example demon-strates the nonlinear and unsteady evolution of a bichromatic wave train with car-rier and riding components of frequency relation f 1=f 2 ¼ 2. According to thepreceding parameter study on simple trigonometric waves in Section 5, the EMDshould be tentatively capable of distinguishing between carrier and riding wavessince amplitudes of both (fundamental) components are nearly equal, although it isknown that the bichromatic wave train evolves nonlinearly in the flume. Fromcomputations using the above cited theoretical approach, specific deviations aresupposed to occur. It is further known a priori that due to nonlinear interactionprocesses, higher order components become prominent. As these are invoked bythe fundamental carrier and riding waves, their spectral representation is very closeto each other and may possibly be superimposed with its source of origin (carrierand riding wave). Amplitudes of higher order components are only fractions ofthat of the fundamental waves so that the EMD technique might be unable to siftthose components individually as a specific IMF. Again, a principle derived for lin-ear trigonometric functions in Section 5.While subfigures in the first column of Fig. 30 show water surface profiles of the

measured and computed wave trains, those of the second and third columnsalready present calculations carried out through the EMD and expose first andsecond IMFs from measured and computed data series. All subfigures in Fig. 30are given in time domain. Obviously, the measured and computed wave trains cor-relate very well—small phase deviations become more and more prominent as the


Fig. 31. Power spectral densities S(f) (f 1=f 2 ¼ 2).

Fig. 30. Water surface elevations g(t) (f 1=f 2 ¼ 2).


wave train propagates downwards the tank. A close correspondence between

amplitude and frequency (periodicity) of the solid and dashed-dotted lines which

represent measured data series and computed water surface elevation records,

respectively, is approximately given for that specific constellation. The EMD effec-

tively decomposes measured and computed nonlinear wave train into two main

components (IMFs) which carry about 98% of the embedded energy and are conse-

quently shown here—the other IMFs are omitted since they are of minor impor-

tance. The first IMF of all six records is exposed in the second column. Again, no

significant deviation between solid and dashed-dotted lines, hence the measured

and computed data, is detected—identical to the complete wave trains from the

first column. Yet again significant deviations become visible from measurement sta-

tions further down the flume. Moreover, frequency divergence in time domain

from both records comes into light as subfigures in the second column expose,

whereas amplitudes of both components are again time-invariant. IMF #2 in the

third column contains the carrier wave for each wave gauge and computed record

very precisely. These components are recognized to be measured and computed

time-invariant in both domains, amplitude and frequency. In contrast to the riding

wave, no phase deviations are evident for the carrier component.

Fig. 32. Hilbert–Huang spectral representation (f 1=f 2 ¼ 2).


From this simple observation, it can be easily proven that the EMD separatescarrier and riding waves very effectively on an adaptive, not a priori known basisalthough frequency relation between both components is small and certain non-linear effects emerge from the signal. It subdivides the measured and computedrecords into a high frequency component mix (IMF #1) and a fundamental carrierwave (IMF #2). The rest of the signal embedded in further IMF is insignificant forthe analysis of the data. According to the preceding Section 5, the EMD isincapable of disintegrating high order nonlinear components on its own which areclose in frequency. Indeed, it is therefore known that all higher nonlinear compo-nents are added to the fundamental riding wave. This assumption can be mostlyconfirmed by analysis results from Fig. 31. In straight correspondence to each mea-sured and computed record from Fig. 30, time-invariant power density spectra(PDS) based on ordinary Fourier technique are presented. Subfigures in the firstcolumn convey measured (solid lines) and computed (thin circles) spectral data.Evidently, second order irregular wave theory turns out to be an appropriate toolto exactly describe dispersive bichromatic nonlinear recorded water waves. Funda-mental components (f1 and f2) are most dominant within the spectra from thewhole data series in the first column of Fig. 31. Bounded irregular superharmoniccomponent B+ governs the spectra as well, but whether the irregular subharmonicB� within all six recorded signals is similarly influential cannot be proven since it is(from theory) of the same frequency (f 2 � f 1) as the carrier wave (f1). Also thedominance of the first superharmonic component is unspecified for this time-invariant spectral representation, because it is of the same frequency (2f1) as theriding wave (f2). Furthermore, the riding waves first bounded superharmonic isclearly evident in the spectra, but generally demands only marginal energy frac-tions. Interacting fundamental and nonlinear higher order components definitelyamplifies nonlinearity of the wave trains on the whole. It is noteworthy that withineach spectra from the measured data, a certain amount of energy is shifted towardslower frequencies. This infragravity wave is neither detected from the wave gaugerecords nor forecast by numerical simulations. Fourier-based spectra of both IMFsare most interesting and principally prove observations previously made fromFig. 30. Again, almost no difference between computed and measured data isexposed. Spectral representations of the first IMF show a broad frequency bandfocused around the riding wave component that embeds major energy fractions.Less dominant are both irregular superharmonic B� and carrier wave which are ofthe same frequency f1 (¼ f 2 � f 1). Nearly the same energy portion is drawntowards the irregular superharmonic component B+. First order bounded har-monic of riding wave is apparent, but of insignificant magnitude. Infragravitywaves are evident for all spectra from the first IMF, although not detected fromthe spectra in the first column representing the full records. The EMD numericallyadds low frequency noise towards its IMF. The second IMF is predominantlyruled by the carrier wave. This component of the bichromatic wave train is repre-sented within a very narrow frequency band. Other spectral fractions are indis-tinguishable, although again some marginal infragravity energy is examined. Onthe whole, the EMD works quite successfully and efficiently for test 1 as both


preceding figures prove. Bichromatic waves are disintegrated effectively, althoughnonlinear components are not directly exposed by the technique.Yet Fig. 32 provides even more detailed information about the disintegrated

components from the EMD as results from the corresponding Hilbert spectralanalysis are presented. Again, 18 subfigures are included. While subfigures in thefirst column of Fig. 32 show water surface profiles of measured and computedwave trains previously documented in Fig. 30, those of the second and third col-umns reveal the Hilbert spectra carried out for each IMF disintegrated from thepreprocessing EMD. Instantaneous frequencies are plotted versus time for eachcharacteristic mode and amplitudes of both measured and the calculated datawhich are shown in time-frequency domain. Gray-scaled color bars adjacent toeach subfigure show corresponding amplitude magnitudes. To distinctly differen-tiate between recorded and numerically simulated data series, the signal’s coordi-nate origin is vertically shifted to establish an individual reference frame. While theupper data series in each subfigure shows the Hilbert transformated IMF of thecalculated bichromatic wave train, the lower one always represents the one fromthe measured signal. Obviously, only marginal dissimilarities between computedand recorded data are evident from the second column in Fig. 32, and hint at thefact that in this particular spectral representation, both signals illustrate an almostidentical internal structure in time-frequency spectral representation. The instan-taneous frequencies of the first IMF, which are assumed to embed the riding waveincluding major fractions of higher order components, exhibit an unusual result.On the one hand, the components’ frequency changes over time in strict periodicityas the time series from which it originates does, but is at the same time, stronglycorrelated with the water surface elevation. Time-variant high and low frequenciesshow an apparent affinity to the wave crests and troughs; consequently, highinstantaneous frequencies correlate with large water surface elevations of the wavesand vice versa for low frequencies. This nonlinear and non-stationary behaviour ofan evolving frequency is rather unfamiliar since researchers in ocean and coastalengineering are used to interpret Fourier-based spectra only. In this representation,information is cancelled out any time so that time-dependent spectra of nonlinearwater waves are somewhat out of the ordinary. Undoubtedly, this extraordinarybehaviour is yet to be described with common perturbation expansion approaches,here, the irregular Stokes wave theory up to the second order. Nonlinear waves arenormally treated by summing up each fundamental constituent with its individualsuperharmonic components. Irregular interaction components are subsequentlyadded. It is concluded that time-variant Hilbert spectra representation unveils theseeffects as a total summation of all nonlinear components including the fundamentalriding wave in one individual IMF. This behaviour has been previously reportedby Schlurmann et al. (2002) for the case of wave trains passing an artificial reefwhich are decomposed into a definite number of individual waves propagatingbehind the structure. Here, the Hilbert spectral analysis for the second IMF con-taining the carrier wave is given in subfigures of the third column of Fig. 32. Time-dependency is neither evident for the numerical simulated nor for the recordeddata series. Instantaneous frequencies are time-invariant and hint at the fact that


the carrier wave is of constant shape over the whole time span data were recorded.The EMD effectively disintegrates time-invariant carrier wave component from thesuperimposed bichromatic wave. Neither bounded superharmonic nor irregularsub- and superharmonics are evident from these spectra.Results from test 2 are given in Figs. 33–35. According to our empirical rules

given in Section 5, the EMD is supposed to work even more profitably as the fre-quency relation increases to f 2=f 1 ¼ 3. Nonetheless, subfigures in the first columnof Fig. 33 show evidence of an inappropriate numerical simulation. Performance isnot as good as test 1 since phase and amplitude deviations are apparent andremind of the fact that the bichromatic wave train gets modified in the wave flume.This unveils the physical effect of time-dependency and indirectly describes thelimitation of a successful numerical description of nonlinear water waves on thebasis of perturbation expansion technique. However, all IMFs in the second andthird columns show similar tendencies, although certain discrepancies betweenmeasured and computed data are uncovered. Besides lacking correspondence intime domain, the EMD disintegrates the signal into two main groups. The first oneshows a broad banded frequency band located around the riding wave (f2), and thesecond part encloses the carrier component (f1), now with a frequency alternatingbehaviour in time domain. Evidently, most of the disclosed carrier waves are modi-fied by some parasitic wave components. The first IMF also shows small tendenciesof time-variant frequencies and amplitudes—Fourier-based analysis is categorically



incompetent to explain the latter effect although most other effects are proven fromthe spectral representations given in Fig. 34. Besides the domination of fundamen-tal waves, each subfigure in the first column shows very pronounced super- andsubharmonic irregular components B+ and B�; also the latter is superimposed withthe first harmonic of the carrier wave as they are of the same frequency. The firstharmonic of the riding wave is apparent, but of negligible influence. The spectralimages of both IMFs demonstrate already known effects from test 1. While the firstis broad-banded including almost every nonlinear irregular component and focusedaround the riding wave, the second is narrow-banded located precisely at the car-rier waves’ frequency, although small fractions are noticeable at the frequency ofthe first harmonic (2f1) and subharmonic irregular component (f 2 � f 1). As pre-viously noticed, infragravity wave energy is evident in both spectral representationsof IMFs, irrespective of whether its origin is measured or computed wave train.Fig. 35 again provides details about the disintegrated components from the

EMD as results from the corresponding Hilbert spectral analysis. Instantaneousfrequencies are plotted versus time for each characteristic mode. Amplitudes ofboth measured and calculated data are shown in time-frequency domain accordingto the adjacent color bars. Data from the second IMF present time-varying fre-quency and amplitude in contrast to the latter experimental configuration, a factthat reminds of the stronger nonlinearity for this bichromatic wave. In summary,the EMD performs quite successfully on test 2, but as it separates carrier and rid-ing waves within the numerical procedure of the EMD, it is unqualified to



distinguish between nonlinear harmonic, irregular sub- and superharmonic andfundamental components on its own. A spectral resolution of the Fourier analysisaccomplishes this HHT restriction and is therefore—at first glance—a more appro-priate tool to deal with nonlinear water waves, but only as long as it deals withtime-invariant signals. However, neither the riding nor the carrier wave componentis time-invariant.Results of the third experimental configuration (test 3) are given in Figs. 36–38.

The presentation and layout are the same for these figures. Some results can bemade from this experiment compared to the preceding studies, although as alreadymentioned, specific disturbances of the bichromatic wave train become more andmore pronounced. The further the superimposed components propagate from itspoint of origin, the more the instability effects arise and heavily modify the pre-sumed harmonic evolution of the wave train. More to the point made earlier, theriding wave component (f 2 ¼ 2:0 Hz) propagates very slowly, so that it is about toreach the last of the six wave gauges after 30 s being generated at the wave board.On the other hand, these disturbances are well known in literature for narrow-ban-ded spectra and are named the Benjamin–Feir instability effects (Benjamin andFeir, 1967). In this context, it has been recognized that the frequency downshift ina wave field evolution is due to nonlinear wave-wave interaction processes. Yet,



detailed mechanisms for these effects are still unknown, and the question arises

whether those instabilities are continuous or gradual. This idea was introduced by

Phillips (1960) who discovered that, in addition to the nonlinear effects bounded

harmonic distortions, there were even weaker nonlinear interactions among differ-

ent free wave components. Although this new nonlinear effect is an order of magni-

tude weaker than the self-interaction, its consequences can be observed through an

accumulation over the span of hundreds of wave periods in time, or hundreds of

wavelengths in spatial domain. Thus, the new nonlinear effect really governs the

instability of the commonly known Stokes waves, the modulation of the wave

envelope, and in fact, all the wave field evolution processes. Consequently, an alter-

native way to study the wave evolution is through the establishment of the envel-

ope function that has been discovered by Zakharov (1968) as Huang et al. (1996)

point out. The governing equation for the envelope was found to be the nonlinear

Schrodinger equation. The presentation of this attempt is far from the scope of the

present paper.A spectral representation of test 3 in Fig. 37 explicitly reveals all previously

named nonlinear components. Besides the fundamental carrier and riding waves,

both bounded first harmonics (2f1, 2f2) and irregular sub- and superharmonic com-

ponents (f 2 � f 1, f 2 þ f 1) are apparent. Theoretical descriptions are appropriate as

long as the bichromatic wave train is near its point of origin (at the wave board).

As it propagates along the tank, strong modulation effects take place and



consequently lead to a distinct mismatch between measurements and computations.The spectra of both IMF replicate observations previously made in tests 1 and 2.The first IMF is broad banded, focused around the carrier wave componentincluding its own bounded first harmonic and irregular sub- and superharmoniccomponents, while the second IMF predominantly represents the carrier waveincluding its—from theory re-known—bounded first harmonic. Clearly, both IMFsslightly overlap in the frequency band. An approximate cut-off frequency for bothIMFs is around 1 Hz. As examined previously, disintegrated wave trains containsmall fractions of low frequency noise that is neither detected from the measureddata sets nor from computations. It is therefore believed to be an outcome of anumerical artifact of the EMD technique. Further studies on that issue have to beperformed in future. Dissimilarities between computed and recorded data are mostevident in the second column in Fig. 38 and again hint at the fact that in this parti-cular spectral representation both illustrate significant differences in the internalstructure that cannot be described with ordinary Stokes approach.

7. Discussion, conclusion and outlook

The present paper relates to the newly developed HHT. A general overview onthis time-frequency analysis technique and its recent applications are given. Thenumerical procedure of the key element, the so-called EMD, is made clear and theprinciple of the HT is introduced. In this context, importance is laid on the cubic



spline interpolation, especially on the algorithms, including its theoretical back-

ground since this parameter plays an essential role when decomposing data series

into IMFs—time-variant narrow-banded components embedded inside the signal.

Furthermore, particular end conditions of the interpolation routines are significant

besides getting to know how to fix the cubic splines to additional characteristic

points of the data series to get the best possible approximation within the iterative

algorithm. The authors wish to stress that all points made in this part of the paper

do certainly need verification and further technical improvements. However, its sig-

nificance and need for development is outlined and should encourage closer investi-

gation by other researchers working in this field.Next, a simple parameter study with trigonometric functions is carried out: a

pragmatic investigation is performed to get an idea about the numerical perform-

ance of the EMD on linear harmonic signals in general. The main results are

shown to estimate relative standardized errors made between analytically exact

defined sine waves and disintegrated intrinsic functions. Their specific influence on

each other is determined. Subsequently, limitations and functional boundaries of

the EMD are derived from this analysis. Note that only two time-invariant sine

wave components have been used for this study. It is assumed that the principle



results are to be transferred to more complex and even time-invariant data sets inclose agreement. Consequences from a simple parameter study are used to investi-gate data sets for practical applications. Here, the HHT is mainly applied to evalu-ate (i) computed nonlinear irregular water waves based on Stokes perturbationexpansion approach and (ii) measurements on fully nonlinear irregular water wavesrecorded in a laboratory wave flume. Excellent correspondence between simulatedand recorded wave trains is given as long as both underlying fundamental compo-nents are close to each other, but do significantly differ (highly nonlinear modifica-tions) as carrier and riding waves get separated. Time-dependent spectralrepresentation shows signs of an interesting phenomenon: instantaneous fre-quencies and amplitudes in the Hilbert spectra exhibit strong correlation withwater surface elevations of both numerical and measured data series.

Acknowledgements

This study was performed within the framework of an extensive research project‘‘On the generating mechanism of transient water from the application of theHilbert transformation technique’’ (SCHL503/5-1) funded by the Deutsche For-schungsgemeinschaft (DFG). In this context, the present article summarizes theessential part of the first half of the research project. The financial support is grate-fully acknowledged by the authors.

References

Bendat, J.S., Piersol, A.C., 1986. Random Data: Analysis and Measurement Procedures. John Wiley &

Sons Inc.

Benjamin, T.B., Feir, J.E., 1967. The disintegration of wave trains on deep water. I. Theory. Journal of

Fluid Mechanics 27, 417–430.

Boashash, B., 1992. Estimating and interpreting the instantaneous frequency of a signal—Part I:

fundamentals. Proceedings of IEEE 80 (4), 520–538.

Braun, S., Feldmann, M., 1997. Time-frequency characteristics of nonlinear systems. Mechanical

Systems and Signal Processing 11 (4), 611–620.

Bronstein, I.N., 1999. Taschenbuch der Mathematik. Harri Deutsch Verlag, (Auflage 4).

Chen, C.-H., Li, C.-P., Teng, L.-T., 2002. Surface-wave dispersion measurements using the

Hilbert–Huang transformation. Journal of Terrestrial, Atmospheric and Ocean Science (TAO) 13

(2), 171–184.

Chien, H., Chuang, L., Gao, C.C., 1999. A study on mechanisms of near-shore rapid waves. In: 2nd

German Clin. J. Seminar on Coastal Eng. Proc., Tainan, China, pp. 469–483.

Cohen, L., Loughlin, P., Vakman, D., 1999. On an ambiguity in the definition of the amplitude and

phase of a signal. Signal Processing 79, 301–307.

de Boor, C., 1978. A Practical Guide to Splines. Springer Verlag.

de Boor, C., 1998. Spline Toolbox Users Guide. The Math Works Inc.

Feldmann, M., 1997. Nonlinear free vibration identification via the Hilbert transform. Journal of Sound

and Vibration 208 (3), 475–489.

Goring, D.G., 2002. Response of New Zealand waters to the Peru tsunami of 23 June 2001. New Zealand

Journal of Marine and Freshwater Research, The Royal Society of New Zealand 36, 225–232.


Gravier, B.M., Napal, N.J., Pelstring, J.A., Jordan, D.A., Miksad, R.W., 2002. An Assessment of the

Application of the Hilbert Spectrum to the Fatigue Analysis of Marine Risers. In: Proceedings of the

11th International Offshore and Polar Engineering Conference, Stavanger, Norway, vol. 2, pp. 268–275.

Hofmann, U.G., Weinhold, J.O., Schrader, P., 2001. Application of empirical mode decomposition and

Hilbert-transformation to multisite neuronal data. In: Poster and extended abstract presented during

10th Annual Computational Neuroscience Meeting, Monterey, CA, USA, (Full paper to appear in

Neurocomputing, 2002).

Huang, N.E., Long, S.R., Sheng, Z., 1996. The mechanism for frequency downshift in nonlinear wave

evolution. Advances in Applied Mechanics 32, 59–120.

Huang, N.E., Shen, Z., Long, S., Wu, M.C., Shih, H.H., Zheng, Q., Yen, N.-C., Tung, C.C., Liu, H.H.,

1998. The empirical mode decomposition and Hilbert spectrum for nonlinear and non-stationary

time series analysis. Proceedings of Royal Society London A 454, 903–995.

Huang, N.E., Shen, Z., Long, S., 1999. A new view of nonlinear water waves: the Hilbert spectrum.

Annual Review of Fluid Mechanics 31, 417–457.

Huang, N.E., Shih, H.H., Shen, Z., Long, S., Fan, K.L., 2000. The ages of large amplitude coastal sei-

ches on the Caribbean Coast of Puerto Rico. Journal of Physical Oceanography 30 (8), 2001–2012.

Huang, N.E., Chern, C.C., Huang, K., Salvino, L.W., Long, S., Fan, K.L., 2001. A new spectral rep-

resentation of earthquake data: Hilbert spectral analysis of Station TCU129, Chi-Chi, Taiwan, 21

September 1999. Bulletin of the Seismological Society of America 91 (5), 1310–1338.

Kuchi, P., Koch, J., 2002. Human gait analysis using the empirical mode decomposition. Technical

Report, Department of Electrical Engineering, College of Engineering and Applied Science, EEE

598C.

Liang, H., Lin, Z., McCallum, R.W., 2001. Artifact reduction in electrogastrograms based on the

empirical mode decomposition. Medical and Biological Engineering and Computing 38 (1), 35–41.

Liu, P.L., 2000a. Is the wind wave frequency spectrum outdated. Journal of Ocean Engineering 27 (5),

577–588.

Liu, P.L., 2000b. Wave grouping characteristics in nearshore Great Lakes. Journal of Ocean Engineer-

ing 27 (11), 1221–1230.

Liu, P.L., 2000c. Wavelet transform and new perspective on coastal and oceanic engineering data analy-

sis. In: Advances in Coastal and Ocean Engineering Book Series, vol. 6. World Scientific.

Long, S.R., Huang, N.E., Xiang, D., 2002. Further enhancements in the use of EMD/HHT in image

analysis. In: Proceedings of the 6th World Multiconference on Systematics, Cybernetics and Infor-

matics (SCI 2002), Orlando, USA, vol. XIV, pp. 276–280.

Longuet-Higgins, M.S., Steward, R.W., 1960. Changes in the form of short gravity waves on long waves

and tidal currents. Journal of Fluid Mechanics 8, 565–583.

Magrin-Chagnolleau, I., Baraniuk, R.G., 1999. Empirical mode decomposition based time-frequency

attributes. In: Proceedings of the 69th SEG Meeting, Houston, TX, USA.

Meeson, R.N., 2001. Non-uniform sampling in real-time within the Hilbert–Huang transform sifting. In:

Proceedings of the 5th World Multiconference on Systematics, Cybernetics and Informatics

(SCI 2001), Orlando, USA, vol. XIV, pp. 281–286.

Meyer, Y., 1993. Book review: ‘‘An introduction to wavelets’’ by C.K. Chui and ‘‘Ten lectures on wave-

lets’’ by I. Daubechies. Bulletin of the American Mathematical Society 28 (2), 350–360.

Oonincx, P.J., 2002. Empirical mode decomposition: a new tool for S-wave detection. Report PNA-

R0203. Probability, Networks and Algorithms (PNA), Centrum voor Wiskunde en Informatica,

National Research Institute for Mathematics and Computer Science, Amsterdam, The Netherlands.

Phillips, O.M., 1960. On the dynamics of unsteady gravity waves of finite amplitude. Part I. Journal of

Fluid Mechanics 9, 193–217.

Pinzon, J.E., 2002. Using the HHT to successfully uncouple seasonal and interannual components. In:

Proceedings of the 6th World Multiconference on Systematics, Cybernetics and Informatics

(SCI 2002), Orlando, USA, vol. XIV, pp. 287–292.

Qiang, G., Xiaojiang, M., Haiyong, Z., Yankun, Z., 2001. Processing time-varying signals by a new

method. Proceedings of the International Conference on Radar (CIE 2001), pp. 1011–1014.


Salisbury, J.I., Wimbush, M., 2002. Using modern time series analysis techniques to predict ENSO

events from the SOI time series. Nonlinear Processes in Geophysics, European Geophysical Society

9, 341–345.

Salvino, L.W., Pines, D.J., 2001. Evaluation of structural response and damping using the empirical

mode decomposition. In: Proceedings of the 5th World Multiconference on Systematics, Cybernetics

and Informatics (SCI 2001), Orlando, USA.

Schaffer, H.A., 1996. Second-order wavemaker theory for irregular waves. Ocean Engineering 23 (1),

47–88.

Schlurmann, T., 2000. The empirical mode decomposition and the Hilbert spectra to analyze embedded

characteristic oscillations of extreme waves. In: Olagnon, M., Athanassoulis, G. (Eds.), Rogue

Waves, pp. 157–165, (Editions Lfremer, ISBN 2-84433-063-0).

Schlurmann, T., 2002. Spectral frequency analysis of nonlinear water waves derived from the Hilbert–

Huang transformation. Journal of Offshore Mechanics and Arctic Engineering (JOMAE), American

Society of Mechanical Engineers (ASME) 124 (1), 22–27.

Schlurmann, T., Graw, K.-U., Lengricht, J., 2000. Spatial evolution of laboratory generated Freak

waves in deep water depth. In: Proceedings of the 10th International Offshore and Polar Engineering

Conference (ISOPE 2000), vol. 3, pp. 54–59.

Schlurmann, T., Dose, T., Schimmels, S., 2001. Characteristic modes of the ‘Adreanov Tsunami’ based

on the Hilbert–Huang transformation. In: Proceedings of the 4th International Symposium on Ocean

Wave Measurement and Analysis (WAVES 2001), American Society of Civil Engineers (ASCE),

vol. 2, pp. 1525–1534.

Schlurmann, T., Bleck, M., Oumeraci, H., 2002. Wave Transformation at Artificial Reefs described by

the Hilbert–Huang Transformation. In: Proceedings of the 28th International Conference on Coastal

Engineering (ICCE 2002), vol. 2, pp. 1791–1803.

Tabor, K., Pinzon, J.E., Brown, M., Tucker, C.J., Myneni, R.B. Relationship between tropical sea sur-

face temperature oscillations and vegetation dynamics in Northern Brazil during 1981 to 2000.

Transaction on Geoscience and Remote Sensing, IEEE, submitted for publication.

Titchmarsh, E.C., 1948. Introduction to the Theory of Fourier Integrals. Oxford University Press.

Veltcheva, A.D., 2002. Wave and group transformation by a Hilbert spectrum. Coastal Engineering

Journal 44 (4), 283–300.

Ville, J., 1948. Theorie et applications de la notion de signal analytic. In: Cables et Transmissions,

Vol. 2a, pp. 61–74.

Vincent, B., Hu, J., Hou, Z., 1999. Damage detection using the empirical mode decomposition method

and a comparison with wavelet analysis. In: Proceedings of the 2nd International Workshop on

Structural Health Monitoring, Stanford, CA, USA, pp. 891–900.

Weisstein, E.W., 2001. Cubic splines. Eric Weisstein’s World of Mathematics. Available from http://

mathworld.wolfram.com/Isometry.html.

Zakharov, J.E., 1968. Stability of periodic waves of finite amplitude on the surface of a deep fluid.

Journal of Applied Tech. Physics 9, 86–94.

Zhang, J., Chen, L., Randall, R.E., 1996. Hybrid wave model for unidirectional irregular waves—part I.

Theory and numerical scheme. Applied Ocean Research 18, 72–92.

Zhu, X., Shen, Z., Eckermann, S.D., Bittner, M., Hirota, I., Yee, J.-H., 1997. Gravity wave character-

istics in the middle atmosphere derived from the empirical mode decomposition method. Journal of

Geophysical Research 102 (D14), 16545–16561.

Zimmermann, C.-A., Seymour, R., 2002. Detection of breaking in a deep water wave record. Journal of

Waterway, Port, Coastal and Ocean Engineering, American Society of Civil Engineers (ASCE) 128

(2), 72–79.

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